Why Arrow’s theorem is a scam. February 17, 2008Posted by Ben Webster in scams.
Since I seem to be on a bit of a social choice kick, let me expound a bit on my pet peeve in social choice theory, the misuse of Arrow’s theorem.
The usual statement of Arrow’s theorem goes something like this: Any voting system which is Pareto efficient and satisfies independence of irrelevant alternatives is a dictatorship.
Pareto efficiency is an extremely weak condition. It simply says that if everyone who votes prefers A to B, B cannot win (certainly any reasonable voting system satisfies this).
Independence of irrelevant alternatives is more touchy. A more accurate name might be the “impossibility of spoilers.” It requires that adding new choices (i.e. candidates) should not change the relative finishing position of the original choices.
IIA is a bit difficult to think about because no voting system you are likely to have heard of satisfies it (by Arrow’s theorem). For example, in the American presidential election, adding a new candidate (let’s call him N) could easily change the outcome between two other candidates (which one might call B and G), for example if G would win a two-way vote with B, but if N runs, he will get the votes of more people who prefer G to B, he can throw the election to B (not that I’m claiming anything like this has ever happened).
Now, spoilers in elections are an upsetting phenomenon (especially for Democrats over the last 8 years), and it would really be nice if we could avoid them. Unfortunately, Arrow’s theorem says that we can’t. Right?
Here’s an examples of a voting system which seems to conflict with Arrow’s theorem:
Rank voting. In this system, each voter gives each candidate a numerical score on some scale (say 1-10). The voter’s scores for each candidate are summed, and the candidate with the highest sum (alternatively, average) wins.
One particularly simple version of this is approval voting, where the options are “approval” (1) or “disapproval” (0).
This system is obviously Pareto efficient and satisfies IIA (changing the score of any candidate won’t affect the relative placement of two others). So how can it be consistent with Arrow’s theorem?
Well, implicit in Arrow’s theorem is a definition of what a “voting system” is, and Arrow’s theorem is based on a rather restrictive (some might say “stupid”) choice. The only acceptable input of a voting system in the Arrow’s theorem schema is an ordered list with no way of measuring the intensities of support for a candidate. So, as far as Arrow is concerned, approval voting isn’t a voting system.
So, give a moment of thought to that the next time somebody claims, well, pretty much anything is a consequence of Arrow’s theorem. And for God’s sake, promote approval voting. It’s sick that people are so excited about instant runoff voting, when approval voting is so obviously superior.